Hyperboloid of One Sheet
The hyperboloid Of One Sheet is a surface of revolution of the curve family hyperbola
![hyperboloid 1 C2xyn](hyperboloid_1_C2xyn.png)
ParametricPlot3D[{Cosh[u]*Cos[v], Cosh[u]*Sin[v], Sinh[u]}, {u, -2, 2}, {v, 0, 2*Pi}, PlotPoints -> 30, MaxRecursion -> 0, Mesh -> All, Boxed -> True, Axes -> False, BoundaryStyle -> {Thin, Black}, MeshStyle -> {Gray}, PlotStyle -> Directive[RandomColor[], Opacity[0.5], Specularity[1, 20]]]
Ruled Surface
A revolving around its transverse axis forms a surface called “hyperboloid of one sheet”. A hyperboloid is a Ruled Surface.
Ruled surfaces are surfaces that for every point on the surface, there is a line on the surface passing it. Or, in other words, a surface generated by a line. If for each point on the surface there are two lines on the surface passing it, then it's called doubly-ruled surface. Hyperboloid is a doubly-ruled surface.
Ruled surfaces also include cylinder and helicoid. There are only 3 doubly ruled surfaces: • Hyperboloid of One Sheet • Hyperbolic Paraboloid • plane.
![hyperboloid 1 sheet gCC9K](hyperboloid_1_sheet_gCC9K.png)
![hyperboloid 1 sheet 7xdwv](hyperboloid_1_sheet_7xdwv.png)
![hyperboloid 1 sheet sJhPY](hyperboloid_1_sheet_sJhPY.png)
![hyperboloid 1 sheet JMTtp](hyperboloid_1_sheet_JMTtp.png)
barLength = 2; offSet = 0.5; girth = 1; tiltLevel = 45*Degree; numberOfTubes = 30; oneTube = Rotate[Tube[{{0, 0, -barLength + offSet}, {0, 0, barLength + offSet}}], tiltX, {1, 0, 0}]; Table[Graphics3D[{RandomColor[], Specularity[1, 20], Table[Rotate[Translate[oneTube, {girth, 0, 0}], n*Degree, {0, 0, 1}], {n, 0, 360, 360/numberOfTubes}]}, Boxed -> False, Axes -> False, ViewPoint -> {2, 0, 1}, PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}], {tiltX, 10*Degree, 70*Degree - 10*Degree, (70*Degree - 10*Degree)/4}]
![hyperboloid sculpture](_p/hyperboloid_metal_sculpture.jpg)
![hyperbolic basket2](_p/hyperbolic_basket2.jpg)
Spinning Cube Silhouette
The silhouette of a rotating dice is a hyperbola.
![spinning cube](_p/spinning_dice.jpg)
![spinning dice photo](rotate_cube.jpg)
![polyhedra dices](polyhedra_dices.jpg)
Hyperboloid in Architecture
![James S McDonnell Planetarium](_p/hypplanet_lg.jpg)
![Kobe port tower d75b4-s500](_p/Kobe_port_tower_d75b4-s500.jpg)
For more photos, see: Hyperboloid Architecture
![hypoid](_p/hypoid.png)